Problem 45 - College Algebra with Corequisite Support

Question

For the following exercises, find all complex solutions (real and non-real). $$ 2 x^{3}-3 x^{2}+32 x+17=0 $$

Step-by-Step Solution

Answer

Use synthetic division, then solve the resultant quadratic polynomial using the quadratic formula to find all roots, real and non-real.

Step 1: Identifying the Nature of the Equation

The given equation is a cubic polynomial, which can have up to three roots. The roots could be real or complex based on the coefficients and the degree of the polynomial.

Step 2: Attempt Rational Root Test

Use the Rational Root Theorem, which states that any rational solution of the polynomial is a factor of the constant term divided by a factor of the leading coefficient. Here, the constant term is 17 and the leading coefficient is 2. Possible rational roots are \( \pm 1, \pm 17, \pm \frac{1}{2}, \pm \frac{17}{2} \). Substitute these into the equation to test if any are roots.

Step 3: Substitute Rational Root Candidates

Substitute these potential rational roots into the equation and evaluate to see if they result in zero. After testing, none of the rational numbers from Step 2 satisfy the equation.

Step 4: Apply Synthetic Division

Since no rational roots were found, use synthetic division to find one root, either real or complex, and reduce the cubic into a quadratic polynomial.

Step 5: Factor the Quadratic Polynomial

Once synthetic division gives a quotient polynomial, factor the resulting quadratic polynomial if possible or use the quadratic formula. The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \).

Step 6: Find Complex Roots using the Quadratic Formula

Apply the quadratic formula formed from the factorization or division in the previous step. Calculate the discriminant of the quadratic. If it is negative, the roots will be complex.

Step 7: Combine All Roots

Combine the real or complex root obtained from synthetic division with the roots obtained from solving the quadratic equation to get all the roots of the original cubic polynomial.

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a useful tool in finding potential rational solutions to polynomial equations. According to this theorem, if a polynomial equation has rational roots, those roots must be in the form \( \frac{p}{q} \), where \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient of the polynomial.
This theorem greatly narrows down the possibilities for rational roots, making it easier to test each one by substitution. For example, in the polynomial \( 2x^3 - 3x^2 + 32x + 17 \), the constant term is 17 and the leading coefficient is 2. Therefore, the possible rational roots are the combinations of the factors of 17 (\( \pm 1, \pm 17\)) divided by the factors of 2 (\( \pm 1, \pm \frac{1}{2} \)), leading to the candidates \( \pm 1, \pm 17, \pm \frac{1}{2}, \pm \frac{17}{2} \).
Each of these candidates can then be substituted into the polynomial to see if they result in zero, indicating they are indeed a rational root.

Cubic Polynomial

A cubic polynomial is a polynomial of degree three, meaning its highest power of \( x \) is \( x^3 \). It can have up to three roots, which can be real or complex. The general form of a cubic polynomial is \( ax^3 + bx^2 + cx + d = 0 \).
In our example, the polynomial \( 2x^3 - 3x^2 + 32x + 17 = 0 \) is a cubic polynomial. Solving it involves identifying whether these roots are real or complex, and it often requires employing various algebraic techniques such as the Rational Root Theorem and synthetic division.
Cubic polynomials can appear challenging, but breaking them down methodically using these techniques can simplify the problem, allowing for a step-by-step approach to finding all roots.

Synthetic Division

Synthetic division is a streamlined method of division that is especially useful for dividing polynomials by linear expressions. It simplifies the traditional polynomial division and involves fewer steps.
To use synthetic division, write down the coefficients of the polynomial followed by the potential root. Perform a systematic process of multiplying and adding to reach a quotient and remainder.
For example, once no rational roots are found for our polynomial \( 2x^3 - 3x^2 + 32x + 17 \), synthetic division helps to reduce it by dividing by a potential factor found earlier or through trial. The output of synthetic division, if the remainder is zero, will be a smaller-degree polynomial, usually a quadratic, which is easier to solve further using additional methods like the quadratic formula.

Quadratic Formula

The quadratic formula is a fundamental tool for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). It provides a direct route to the roots by using the formula:\[ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \]The part under the square root, \( b^2 - 4ac \), is called the discriminant and determines the nature of the roots:

• If the discriminant is positive, the quadratic has two distinct real roots.
• If the discriminant is zero, it has exactly one real root, sometimes called a repeated root.
• If the discriminant is negative, the quadratic has two complex roots.

After reducing the original cubic polynomial by finding one root through synthetic division, the resulting quotient is typically a quadratic equation, which can then be solved using the quadratic formula. In our example, this approach reveals the remaining roots, whether real or complex, of the polynomial. The quadratic formula is especially useful when factoring is not straightforward.